In practice, the minimum uncertainty in δ 1 is likely larger than the sensor resolution due to other factors (e.g., background fluctuations, drifts in temperature). This limit could be increased by reducing probe spacing or increasing the size of the heat pulse. However, closer spacing magnifies relative errors caused by imperfect alignment of probes during installation.
The CHPM was unable to detect high sap velocities because, like the HRM, it is limited by uncertainty in δ 1. The CHPM also fails at low sap velocities because the intrinsic uncertainty in the ”crossover point” of δ 1 andδ 3 is greater. Both the CHPM and the Tmax methods are inherently incapable of detecting negative sap velocities (reverse flows). Testi and Villalobos (2009) suggested the resolution of the CHPM could be improved using an empirical calibration function (Vandegehuchte & Steppe, 2012c), albeit this requires study and species-specific solutions.
Pearsall et al. (2014) obtained estimates of sap velocity > 200 cm h-1 by supplementing the HRM with the CHPM. Those velocities far exceed values that we were able to detect using the CHPM (~ 40-50 cm h-1). Given that Pearsall et al. (2014) used temperature sensors and dataloggers with similar resolution to ours, and heater probes with similar resistances (18 Ω), this discrepancy in maximum velocities may be due to wider probe spacing used in the present study (1.5 cm vs 1.2 cm used by Pearsall et al. 2014). To assess this possibility, we can use Eqn. 1 to quantify the effect of probe spacing on the maximum temperature rise that one would expect to occur at the proximal probe (Probe #1). Suppose ρ = 1.08⋅103 kg m-3,c = 2.8⋅103 J kg-1K-1, Q = 1370 J m-1 (a 3.5 cm probe with 18 Ω resistance and 6 s pulse at 12 V), and k = 0.0025 cm2 s-1; then, at a true heat pulse velocity of 100 cm h-1 (0.028 cm s-1), δ 1 should reach 0.0099 K for a probe spacing of 1.2 cm, and 0.0014 K for 1.5 cm. This difference results from the exponential dependence of temperature rise on probe spacing (Eqn. 1 ). We suggest that superior performance of the DRM compared to the CHPM at high sap velocities is likely due to wider probe spacing (i.e. compared to that used by Pearsall et al. (2014)). Wider spacing has the advantage that a given absolute error of probe alignment during installation will cause a smaller error in calculated sap velocity.
The DRM has other advantages. Most importantly, the measurement principle (Eqn. 2 ) is the same across all sap velocities. Switching measurement principles, as required if HRM is used for slow sap velocities and the CHPM for fast sap velocities, creates logistic problems and carries the risk of greater variation due to edaphic conditions. Secondly, the DRM provides a theoretical basis for identifying the optimal time window in which to average estimates of sap velocity. We found an optimal window for E. cypellocarpa of between 50 and 120 s (see Fig. S6 ), which is approximately equal to theoretically calculated values (Fig. 4 ) and similar to the window used for the HRM. Thirdly, the DRM is less sensitive to noise than the CHPM, because the latter relies on a single intersection point, whereas the former averages data over a longer time period. Finally, the DRM is relatively insensitive to uncertainty in the value of k when V >20 cm h-1because V DRM = V 23 (Eqn.6 ) at high sap velocities, and the term involving k in Eqn. 6 becomes small compared to the term that involvest .
4.2 Impact of the finite heat pulse length
Following Marshall’s original model, the assumption of an instantaneous heat pulse results in biased estimates of sap velocity in most methods. A small negative bias is evident in both the DRM- and HRM-based velocity estimates. The bias of the DRM increases from -0.03 % when V = 30 cm h-1 to -0.2 % when V = 80 cm h-1 (Fig. 9 ). A larger bias is evident using the CHPM (underestimates velocity by 9 % at 30 cm h-1) because of reliance on the precise timing whereδ 1=δ 3. Its high sensitivity to t0 is also recognized in a data synthesis study of Flo et al. (2019). The Tmax method can be modified to account for t0 (Kluitenberg & Ham, 2004), thus eliminating this bias. In practice, Tmax-based sap velocity estimates are highly uncertain when velocity is high, due to sensitivity to noise (Fig. 5e ). Shiftingt0 helps mitigate the impact of the finite heat pulse length across methods, resulting in negligible bias for the DRM (Fig. 9 ).